I got stuck in the following problem, any help would surely be appreciated:

I have to prove that a block matrix of the form:

is positive definite, where is symmetric and positive definite, and is anti-symmetric, that is, .

When I take the following nonzero vector:

and try to check the usual condition , I get the following:

But:

And is a scalar, so:

Substituting above, we have:

Now, how can I prove that the real number is always positive?

For sure, and are both positive, or at least one of them is positive and the other is zero, because is positive definite.

Thanks,

Henrique

May 30th 2010, 09:58 AM

Opalg

Quote:

Originally Posted by thetouristbr

Hello everyone,

I got stuck in the following problem, any help would surely be appreciated:

I have to prove that a block matrix of the form:

is positive definite, where is symmetric and positive definite, and is anti-symmetric, that is, .

When I take the following nonzero vector:

and try to check the usual condition , I get the following:

That result is false. For example, take 2×2 matrices and . Let and . Then .

If you don't like taking (because it's not strictly speaking positive definite) then you can replace it with a small positive multiple of the 2×2 identity matrix, and you will still find that is negative.

May 30th 2010, 10:26 AM

Opalg

Quote:

Originally Posted by Opalg

That result is false. For example, take 2×2 matrices and . Let and . Then .

If you don't like taking (because it's not strictly speaking positive definite) then you can replace it with a small positive multiple of the 2×2 identity matrix, and you will still find that is negative.

On the other hand, if the scalars are supposed to be real, and if (without the minus sign in the top right corner), then you can check that the result holds. That is probably what was intended.